A115147 Eighth convolution of A115140.

1, -8, 20, -16, 2, 0, 0, 0, -1, -8, -44, -208, -910, -3808, -15504, -62016, -245157, -961400, -3749460, -14567280, -56448210, -218349120, -843621600, -3257112960, -12570420330, -48507033744, -187187399448, -722477682080, -2789279908316, -10772391370048, -41620603020640

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1670

Formula

O.g.f.: 1/c(x)^8 = P(9, x) - x*P(8, x)*c(x) with the o.g.f. c(x) = (1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(9, x) = 1-7*x+15*x^2-10*x^3+x^4 and P(8, x) = 1-6*x+10*x^2-4*x^3.

a(n) = -C8(n-8), n>=8, with C8(n) = A003518(n+3) (eighth convolution of Catalan numbers). a(0) = 1, a(1) = -8, a(2) = 30, a(3) = -16, a(4) = 2, a(5) = a(6) = a(7) = 0. [1, -8, 20, -16, 2] is row n=8 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.

From Amiram Eldar, Oct 09 2025: (Start)

a(n) = -8 * binomial(2*n-9, n-8)/n for n >= 1.

a(n) ~ -4^(n-3) / (n^(3/2) * sqrt(Pi)). (End)

Mathematica

CoefficientList[Series[(1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*x^3) *Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2 -4*x^3)*sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*x^3)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019
(SageMath) ((1-8*x+20*x^2-16*x^3+2*x^4 +(1-6*x+10*x^2-4*x^3)*sqrt(1-4*x))/2 ).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

Crossrefs

Cf. A000108, A003518, A034807, A115139 - A115146, A115148, A115149.

Sequence in context: A208085 A334065 A128909 * A302241 A022700 A214457

Adjacent sequences: A115144 A115145 A115146 * A115148 A115149 A115150

Keyword

sign,easy

Author

Wolfdieter Lang, Jan 13 2006

Status

approved